I.J. Rao

The effect of the slip boundary condition on the flow of fluids in a channel

SummaryThe assumption that a liquid adheres to a solid boundary (“no-slip” boundary condition) is one of the central tenets of the Navier-Stokes theory. However, there are situations wherein this assumption does not hold. In this paper we investigate the consequences of slip at the wall on the flow of a linearly viscous fluid in a channel. Usually, the slip is assumed to depend on the shear stress at the wall. However, a number of experiments suggests that the slip velocity also depends on the normal stress. Thus, we investigate the flow of a linearly viscous fluid when the slip depends on both the shear stress and the normal stress. In regions where the slip velocity depends strongly on the normal stress, the flow field in a channel is not fully developed and rectilinear flow is not possible. Also, it is shown that, in general, traditional methods such as the Mooney method cannot be used for calculating the slip velocity.

A study of strain-induced crystallization of polymers

The response of polymers depends on their morphology. One of the challenges in modeling from a continuum perspective is how to incorporate the microstructural features into the homogenized continuum model. Here, we use a recent framework that associates different natural states and material symmetries with distinct microstructures of the body (Rajagopal, K.R., Wineman, A.S., 1992. International Journal of Plasticity 8, 385; Rajagopal, K.R., 1995. Reports of the Institute for Computational and Applied Mechanics, University of Pittsburgh; Rajagopal, K.R., Srinivasa, A.S., 1995. International Journal of Plasticity 11, 653; Rajagopal, K.R., Srinivasa, A.S., 1999. ZAMP 50, 459). We study the problem of strain-induced crystallization of polymeric materials, in particular, we study the problem of uniaxial stretching of polymeric materials and the subsequent crystallization and the predictions of the theory are compared with experimental results.

Some simple flows of a Johnson-Segalman fluid

SummaryUnlike most other fluid models, the Johnson-Segalman fluid allows for a non-monotonic relationship between the shear stress and rate of shear in a simple shear flow for certain values of the material parameter. This has been used for explaining a phenomenon such as “spurt”. Here, we study three simple flows of a Johnson-Segalman fluid with a view towards understanding its response characteristics. We find that boundary conditions can have a very interesting effect on the regularity of the solution; changing them continuously leads to solutions that change their regularity. First, we consider the flow through a circular pipe and find solutions that have discontinuous velocity profiles which have been used to explain the phenomenon of “spurt” (cf. [10], [11]). Second, we consider the flow past an infinite porous plate and show that it will not admit solutions which have discontinuous velocity gradients, the solutions being necessarity smooth. Lastly, we study Poiseuille flow in a concentric annulus with porous boundaries. While “spurt” could be explained alternatively by allowing for “stick-slip” at the wall, the Johnson-Segalman model seems particularly suited in describing the appearance of “shear-layers” (cf. [13]).

Phenomenological modelling of polymer crystallization using the notion of multiple natural configurations

Crystallization and solidification in polymers is a problem of great importance to the polymer processing industry. In these processes, the melt is subjected to deformation while being cooled into the desired shape. The properties of the final product are strongly influenced by the deformation and thermal histories and the final solid is invariably anisotropic. In this work we present a model to capture the effects during solidification and crystallization in polymers within a purely mechanical setting, using the framework of multiple natural configurations that was introduced recently to study a variety of non-linear dissipative responses of materials undergoing phase transitions. Using this framework we present a consistent method to model the transition from a fluid-like behaviour to a solid-like behaviour. We also present a novel way of incorporating the formation of an anisotropic crystalline phase in the melt. The anisotropy of the crystalline phase, and consequently that of the final solid, depends on the deformation in the melt at the instant of crystallization, a fact that has been known for a long time and has been exploited in polymer processing. The proposed model is tested by solving three homogenous deformations.

A thermomechanical framework for the glass transition phenomenon in certain polymers and its application to fiber spinning

A thermodynamic framework is developed to describe a polymer melt undergoing glass transition that takes into account the fact that during such a process the underlying natural configurations (stress-free states) are continually evolving. Such a framework allows one to take into account changes in the symmetry of the material, if such changes take place. Moreover, the framework allows for a seamless transition of a polymeric melt to a mixture of a melt and an elastic solid to the final purely solid state. The efficacy of the model is tested by studying the fiber spinning problem for polyethylene terephthalate and the predictions agree well with the experimental results.

Flow of a Johnson-Segalman fluid between rotating co-axial cylinders with and without suction

Abstract The Johnson–Segalman fluid has a non-monotone relationship between the shear stress and velocity gradient in simple shear flows for a certain range of material parameters resulting in solutions with discontinuous velocity gradients for planar and cylindrical Poiseuille flow. This has been used to explain the phenomenon of “Spurt”. Rao and Rajagopal [ Acta Mechania , to be published] have shown that the addition of suction necessarily smoothens the solutions for planar Poiseuille and cylindrical Poiseuille flows. Here we study the effect of a different geometry on the flow of a Johnson–Segalman fluid. The problems of cylindrical Couette flow, cylindrical Couette flow with suction (or injection) and Hamel flow are studied and it is found that the boundary condition can have an interesting effect on the regularity of the solution. The presence of suction increases the regularity of the solution, i.e. solutions with discontinuous velocity gradients are not possible.

Effect of the rate of deformation on the crystallization behavior of polymers

Deformation has a significant influence on the crystallization process in a number of polymers. In this paper, the response of a recently developed model for crystallizing polymers is investigated when subject to uni-, bi-axial and constant width extensions for a range of strain rates. Both the loading and unloading behavior are examined for these deformations. The particular model studied here was developed to capture the effect of strain induced crystallization in polymers and has been applied to model crystallization in polyethylene terephthalate at temperatures just above its glass transition temperature. The model has been formulated using the notion of multiple natural configurations within a full thermodynamic framework. The connection between micro-structural changes taking place in the polymer and the form of the model are elucidated. The interplay between the relaxation processes, the rate of deformation and their combined effect on crystallization is illustrated. The results show an earlier onset of crystallization for high strain rates due to stretching of the polymer network. At low strain rates however, crystallization is not observed as the polymer network is able to relax during the deformation. A sharp upturn in the stress is observed after the onset of crystallization due to the formation of a rigid crystalline phase. The unloading curves clearly show a hysteric behavior with the amount of dissipation increasing for increasing values of strain rate. These results compare favorably with experimental observations available in literature.

A Thermodynamic Framework for Describing Solidification of Polymer Melts

A thermodynamic framework is presented that can be used to describe the solidification of polymer melts, both the solidification of atactic polymers into an amorphous elastic solid and the crystallization of other types of polymer melts to semi-crystalline elastic solids. This framework fits into a general structure that has been developed to describe the response of a large class of dissipative bodies. The framework takes into account the fact that the natural configuration of the viscoelastic melt and the solid evolve during the process and that the symmetries of these natural configurations also evolve. Different choices are made as to how the material stores energy, produces entropy, and for its latent heat, latent energy, etc., that lead to models for different classes of materials. The evolution of the natural configuration is dictated by the manner in which entropy is produced, how the energy is stored etc., and it is assumed that the constitutive choices are such that the rate entropy production is maximized, from an allowable class of constitutive models. Such an assumption also determines the crystallization kinetics, i.e., provides equations such as the Avrami equation. Using the framework, a model is developed within which the problem of fiber spinning is studied and we find that the model is able to predict observed experimental results quite well.

Simulation of the Film Blowing Process for Semicrystalline Polymers

Abstract We simulate the film blowing process using a fully thermodynamic model developed to study crystallization in polymers. The model stems from a general thermodynamic framework that is particularly well suited to describing dissipative processes during which the symmetry of the material can change; thus, it provides a good basis for studying the crystallization process in polymers. The polymer melt is modeled as a rate-type viscoelastic fluid and the crystalline solid polymer is modeled as an anisotropic elastic solid. The initiation criterian, marking the onset of crystallization and equations governing the crystallization kinetics, arise naturally in this setting in terms of the appropriate thermodynamic functions. The mixture region, wherein the material transitions from a melt to a semicrystalline solid, is modeled as a mixture of a viscoelastic fluid and an elastic solid. The anisotropy of the crystalline phase and consequently that of the final solid depends on the deformation in the melt during crystallization. The polymer melt is simulated using a rate-type model that is a generalization of the Maxwell model, which allows for the relaxation time to depend on the deformation. The results of the simulation agree qualitatively with experimental observations and the methodology described provides a framework in which the film blowing problem can be analyzed. Most previous attempts at describing film blowing terminate at the point at which the bubble reaches its maximum diameter. In fact, a continuation of the simulation in the previous studies, instead of terminating it abruptly on an ad hoc basis as is commonly done, would predict a collapsing bubble because the transition of the material from a fluid model to a solid model is not accounted for. This inadequacy of previous approaches to studying the problem cannot be overemphasized. Rather than just increasing the viscosity of the fluid, which is the usual practice, a true transition to a solid model is adopted in this study and this leads to a model that captures the smooth transition to stable bubble development.